hmwk9 - x 2 y 2 z 2 = 4 x and inside the paraboloid of...

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Math 215 Homework Set 9: §§ 17.4 – 17.6 Winter 2008 Most of the following problems are modified versions of the recommended homework problems from your text book Multivariable Calculus by James Stewart. 17.4d. Please do Problem 22 of § 17.4 in Stewart’s Multivariable Calculus . 17.4e. Verify Green’s Theorem for the line integral C x 3 y dx - x 2 y 2 dy where C consists of the parabola y = x 2 - 4 from (1 , - 3) to ( - 1 , - 3) and the line segment from ( - 1 , - 3) to (1 , - 3) . 17.5a. The role of curl and divergence in multivariable Calculus is similar to the role of words in a com- position; in order for the final product to make sense, you need to know in which order things may be placed. To this end, please do Problems 12, 19, 20, 23–29 of § 17.5 in Stewart’s Multivariable Calculus . 17.6a. Find the area of the finite part of the paraboloid of revolution z = 3 x 2 + 3 y 2 cut off by the plane z = 75 . 17.6b. Find the area of the surface that lies on the sphere
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Unformatted text preview: x 2 + y 2 + z 2 = 4 x and inside the paraboloid of revolution x = y 2 + z 2 . Hint: Sketch the surface. 17.6c. Find an equation for the tangent plane to the surface parameterized by r ( u, v ) = ± u 2 , v 2 , uv ² at the point (4 , 4 ,-4) . Sketch a graph of the surface and the tangent plane. Use MAPLE if you wish. 17.6d. Please do Problems 13-18 of § 17.6 in Stewart’s Multivariable Calculus . 17.6e. Sketch the surface described by the parameterization r ( u, v ) = ± 2 u cos( v ) , 2 u sin( v ) , v ² for 1 ≤ u ≤ 4 and v ∈ [ π, 2 π ] . Find the area of this surface. 17.6f. Find a parametric representation for the upper half of the ellipsoid 4 x 2 + 2 y 2 + z 2 = 1 . 17.6g. Find a parametric representation for that part of the sphere of radius 12 centered at the origin that lies above the cone z = ² x 2 + y 2 ....
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